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Busy period in GIX/G/∞

Part of: Computer system organization Special processes

Published online by Cambridge University Press:  14 July 2016

Liming Liu  and
Ding-Hua Shi
Liming Liu*
Affiliation:
Hong Kong University of Science and Technology
Ding-Hua Shi*
Affiliation:
Shanghai University of Science and Technology
*
∗Postal address: Department of Industrial Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong.
∗∗Postal address: Department of Mathematics, Shanghai University of Science and Technology, Shanghai, China.
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Abstract

Busy period problems in infinite server queues are studied systematically, starting from the batch service time. General relations are given for the lengths of the busy cycle, busy period and idle period, and for the number of customers served in a busy period. These relations show that the idle period is the most difficult while the busy cycle is the simplest of the four random variables. Renewal arguments are used to derive explicit results for both general and special cases.

Keywords

BUSY PERIODBUSY CYCLEIDLE PERIODVECTOR MARKOV PROCESS METHODRENEWALMETHOD

MSC classification

Primary: 68M20: Performance evaluation; queueing; scheduling
Secondary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 3 , September 1996 , pp. 815 - 829
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] Brown, M. and Ross, S. (1969) Some results for infinite server queues. J. Appl. Prob. 6, 604–611.Google Scholar
[2] Browne, S. and Steele, J. M. (1993) Transient behavior of the coverage processes with applications to the infinite-server queue. J. Appl. Prob. 30, 589–602.Google Scholar
[3] Chaudhry, M. L., Agarwal, M. and Templeton, J. G. C. (1992) Exact and approximate numerical solutions of steady-state distributions arising in the queue GI/G/1. Queueing Systems 10, 105–152.Google Scholar
[4] Dvurecenskij, A. (1984) The busy period of order n in the GI/D/8 queue. J Appl. Prob. 21, 207–212.Google Scholar
[5] Holman, D. F., Chaudhry, M. L. and Kashyap, B. R. K. (1982) On the number in the system GIX/M/8. Sankyä A44, 294–297.Google Scholar
[6] Keilson, J. and Servi, L. (1994) Network of non-homogenous M(t) /G/8 systems. J. Appl. Prob. 31A, 157–168.Google Scholar
[7] Liu, L., Kashyap, B. R. K. and Templeton, J. G. C. (1990) On the GIx/G/8 system. J. Appl. Prob. 27, 671–683.Google Scholar
[8] Liu, L. and Templeton, J. G. C. (1993) Autocorrelations in infinite server batch arrival queues. Queueing Systems 14, 313–337.Google Scholar
[9] Shanbhag, D. N. (1966) On infinite server queues with batch arrivals. J. Appl. Prob. 3, 274–279.Google Scholar
[10] Shi, D. H. (1990) Probability analysis of the repairable queueing system M/G(Ek/H) / 1. Ann. Operat. Res. 24, 185–203.Google Scholar
[11] Stadje, W. (1985) The busy period of the queueing system M/G/8. J. Appl. Prob. 22, 694–704.Google Scholar
[12] Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar